Durch die Nachhilfe ist Mathe mir jetzt viel klarer.

durch means “through” or “because of / as a result of” in this context:
Durch die Nachhilfe ≈ Because of the tutoring / Thanks to the tutoring.

Grammatically, durch always takes the accusative case.

• Base noun: die Nachhilfe (feminine, singular, nominative)
• After durch (accusative): still die Nachhilfe
  (for feminine singular, nominative and accusative look the same)

So the phrase is: durch + die (Akk.) + Nachhilfe.

The order isn’t strictly fixed here. Both are possible:

• Durch die Nachhilfe ist mir Mathe jetzt viel klarer.
• Durch die Nachhilfe ist Mathe mir jetzt viel klarer.

Differences:

• German word order in the middle field (between verb and sentence end) is relatively flexible.
• A common tendency: pronouns (like mir) usually come before full nouns (Mathe), so „ist mir Mathe …“ is probably the most neutral, everyday version.
• In „ist Mathe mir …“, Mathe is placed earlier for emphasis or topic-focus: you’re highlighting Math as the topic (“As for math, it’s clearer to me now”).

So the given sentence is correct, but „ist mir Mathe jetzt viel klarer“ might sound slightly more natural in everyday speech.

German often uses a dative “experiencer” to express what is clear/known/understandable to someone.
With adjectives like klar, bewusst, peinlich, etc., dative is normal:

• Etwas ist mir klar. – Something is clear to me.
• Es ist mir peinlich. – I’m embarrassed by it / It’s embarrassing to me.
• Es ist mir bewusst. – I’m aware of it.

So:

• Mathe ist mir klarer. = Math is clearer *to me.*

Using mich would be wrong here (ist mich klarer doesn’t work).

Using für mich would change the nuance:

• Mathe ist für mich klar. – can mean “Math is clear for me / in my opinion / from my perspective,” less about the internal experience and more about viewpoint, suitability or relevance.

In this sentence, the natural idiom is „etwas ist mir klar(er)“ with dative mir.

Mathe is the subject of the sentence, so it’s in the nominative case.

Structure:

• Mathe – subject (what is clearer?)
• ist – linking verb (form of sein)
• mir – dative experiencer (to me)
• viel klarer – predicate adjective phrase

In German, predicate adjectives after sein (like klar, klarer) describe the subject, so Mathe must be nominative, even though there’s no article to show it.

Mathe is the everyday short form of „Mathematik“, and it means “math” / “mathematics”.

Nuances:

• die Mathematik – full, formal term (used in academic or written contexts)
• Mathe – colloquial/neutral everyday word (used in school, conversations, etc.)

In this sentence, Mathe is used without an article, which is very natural for school subjects:

• Mathe ist schwer. – Math is hard.
• Englisch macht mir Spaß. – I enjoy English.

You could also say „Durch die Nachhilfe ist Mathematik mir jetzt viel klarer“, but it would sound more formal and a bit unusual in casual speech.

klarer is the comparative form of the adjective klar (klar → klarer).

German (like English) usually forms comparatives by adding -er to short adjectives:

• klar → klarer – clear → clearer
• gut → besser – good → better (irregular)
• schnell → schneller – fast → faster

You do not say „mehr klar“ in normal German. Use the comparative form:

• ✓ Mathe ist mir jetzt klarer. – Math is clearer to me now.
• ✗ Mathe ist mir jetzt mehr klar. – incorrect.

So klarer is simply the correct comparative of klar.

Here, viel is an intensifier modifying the comparative klarer:

• klarer – clearer
• viel klarer – much clearer, a lot clearer

In English you can’t usually say “very clearer”, so you say “much clearer” instead.
German does exactly that:

• ein bisschen klarer – a bit clearer
• deutlich klarer – significantly clearer
• viel klarer – much clearer / a lot clearer

In the sentence:

Durch die Nachhilfe ist Mathe mir jetzt viel klarer.

jetzt (“now”) sits between the pronoun mir and the adjective phrase viel klarer. This is a typical and natural position for time adverbs.

Other common options:

• Durch die Nachhilfe ist mir Mathe jetzt viel klarer.
• Durch die Nachhilfe ist mir jetzt Mathe viel klarer. (less common, but possible)
• Jetzt ist mir durch die Nachhilfe Mathe viel klarer. (puts focus on now)

General tendencies:

• Time adverbs often go fairly early in the middle field, but there is flexibility.
• The chosen position „… ist Mathe mir jetzt viel klarer“ sounds very normal and natural; moving jetzt slightly usually doesn’t change the meaning, just the emphasis.

Nachhilfe is more specific than just “help”. It typically means:

• private tutoring
• extra lessons
• remedial instruction / extra help with schoolwork

So Durch die Nachhilfe suggests:

• Through the tutoring I’m getting
• Thanks to the extra help (from a tutor/teacher)

It’s usually about structured, regular help with a school subject, not just a friend casually explaining something once.

Both are possible, but there’s a nuance:

• durch (Akk.) – emphasises means / process / direct cause
  → as a result of going through the tutoring
• wegen (Gen./Dat. in colloquial speech) – emphasises reason / cause
  → because of the tutoring

So:

• Durch die Nachhilfe ist Mathe mir jetzt viel klarer.
  Focus: the tutoring as the means that made math clearer.

• Wegen der Nachhilfe ist Mathe mir jetzt viel klarer.
  Focus: the tutoring as the reason; can sound a bit more neutral and slightly less “instrumental”.

In everyday language, both are acceptable. The original with durch nicely fits the idea “Thanks to the tutoring itself, math is clearer now.”

Yes, you can say:

• Ich verstehe Mathe jetzt viel besser. – I understand math much better now.

Difference in nuance:

• Mathe ist mir (jetzt) viel klarer.
  – Focus on math being clear to you; a state: clarity, transparency.
  – Uses the idiom „etwas ist mir klar“ (something is clear to me).

• Ich verstehe Mathe jetzt viel besser.
  – Focus on your act of understanding, an ability/process: you can comprehend it better now.

In context, they usually mean roughly the same thing, but the original sentence is framed around the clarity of the subject rather than explicitly about your understanding action.